Question

Stuck on a relatively simple question:

dy/dx of y= x/sqrt(1-x^2)?

Answer 1

Here's where I'm at using quotient and chain rule for denominator:
\[y= x / \sqrt{1-x ^{2}}\]
\[dy/dx = (\sqrt{1-x ^{2}} - (1/2 * -2x * \sqrt{1-x ^{2}}^{-1/2}*x )) / (\sqrt{1-x ^{2}})^{2}\]
\[dy/dx = (\sqrt{1-x ^{2}} +x ^{2}*\sqrt{1-x ^{2}}^{-1/2}) / (1-x ^{2})\]

Now somehow this simplifies greatly, I can't figure it out though.

Thanks in advance.

Answer 2

Well, I kept plugging away at it and realized I need to implicitly differentiate this.

I started by squaring both sides and found the answer that way. Is there an easy way to simplify the above expression?

Answer 3

That's a great question Scott. However, a followup question I want to ask you is what is your policy on negative exponents? Negative exponents, for me at least, need to be written as positive exponents which involve putting the expression that is enclosed by exponents either in the denominator or the numerator. As an example: (1-x)^(-1/2) needs to be written as 1/(1-x)^(1/2)

Answer 4

Msdoroff, I go back and forth on how I write negative exponents. I'm on the fence as to what I prefer.

Answer 5

Okay! not a problem. Well, if you divide both terms for square root(1-x^2) you should end up with 1+x^2/square root of 1-x^2. After you get that, you could invert the denominator (which is where you multiply the reciprocal of the denominator and do a little bit more algebra. How simple of an answer are you looking for?

Answer 6

This is just a simpler part of a harder question. The entire question is Calculate the derivative with respect to x for:

\[\tan ^{-1}(x/\sqrt{1-x ^{2}})\]

I played around with my first attempt and if you multiply the top and bottom by
\[\sqrt{1+x ^{2}}\]
it simplifies to
\[1/\sqrt{1-x ^{2}}^{3/2} \]

I needed to figure out the derivative of the inside before I tackled the larger problem, which I'm still struggling with, but I'm getting close

Answer 7

you just use the chain rule. You take the derivative of arctanx(tan^-1) and then you take the derivative of the inside function which is x/square root(1-x^2)

Answer 8

oh! I see where the problem is!

Answer 9

Let me work out a solution to this problem and i'll get back to you.

Answer 10

This is a very easy question.Use trigonometric substitution to simplify the function.The solution is in the picture.

Answer 11

@Scott* Direct differentiation is very hard for this problem.

Answer 12

Thank you very much. I need to really work on finely tuning my trig skills. I have a hard time seeing these types of substitutions. Thanks again.

Answer 13

You're welcome.

Answer 14

I thought trig substitution only worked for integrals....

Answer 15

Trig substitution works at any time you are manipulating trig functions. The circular nature of them allows you to simplify almost any expression involving trig functions (assuming you know what you are doing). I clearly need to work on that.